How do a transform simple object to have a concave shape
I was able to do it in surprisingly few steps using a shrinkwrap modifier.
Here's my version of your starting shape.

I noticed that yours isn't at the worldspace origin. You will want it there for the steps below. Select your headboard in Object Mode, right click on it, select Set Origin, and then choose Geometry, to make sure you're object center is in the middle of your mesh. Then press Shift + c to force the 3D Cursor to the worldspace origin (just in case it isn't there already). This will reorient your view, though that outcome was not intended. Press numpad period to get recentered on your selected object, if necessary. Then press Shift + s to bring up the Snapping menu, and choose Selection to Cursor. Now your object is centered at the Worldspace origin.
Start by creating a mesh cylinder (Add &gt; Mesh &gt; Cylinder)  and take note of the starting options I chose in the Operator panel to the bottom left of the 3D View. You may find that you can't use the same numbers because of a difference in scale between my version of your headboard and your own. I chose 48 because that allowed 12 sides of the cylinder to be placed precisely between the two points of intersection shown in the pic below, at the two red arrows, which is what you want. 12 is the magic number, based on your very own model, because that's the number of sides that designates the row of faces that you want to build into an arch.

Here's what it may look like after you have the new shape in place. Don't worry about how it looks or how long the cylinder is. It doesn't matter.

Next, make sure you're in Object Mode, select the new cylinder you just made, and hide it (h), then select the headboard, and go into Edit Mode. Select all of the vertices shown below. These are the ones you want to change into a new shape. Next you will want to create a vertex group, which will remember this particular selection of vertices, and allow you to target it later in other operations. Create the vertex group by clicking the &quot;+&quot; icon under the Vertex Group Section of the Object Data tab in the Properties Panel. With your vertices selected, press Assign.

Next, go back into Object Mode and, with your headboard selected, go to the Modifiers tab of the Properties Panel and create a Shrinkwrap modifier. Assign the vertex group you created to it. You may also want to unhide your cylinder again (Alt + h in Object Mode) so that you can find it more easily in the next step.

And finally, assign the target, which is the cylinder you created earlier. This will appear to have no effect until you hide the cylinder afterwards. If you want to make this effect permanent, you will need to apply (Ctrl + a) the shrinkwrap modifier. After doing that, you can delete your cylinder. You don't need it anymore.
Hope this helps.